Using the decorations constructed above, we deduce that the contributions of weak and second-order topological phases with topological invariants \(\mathbf {G}_\nu = \left(\nu _x, \nu _y, \nu _z \right)^T\)  and \(\nu _{2 \pi / n}\) 
The topological invariants \(\nu _i, \ i=x,y,z\)  and \(\nu _{2\pi /n}\)  for the weak and second order topological phases, respectively, can be abstractly defined through the isomorphism \(\nu \)  from the abelian group of stable topological equivalence classes of Hamiltonians [1]} to its isomorphic abelian group of integers \(\mathbb {Z}\)  or \(\mathbb {Z}_2\) , with addition as group operation and generator "1". For second-order topological phases, we argued in section  that their classifying group is isomorphic to \(\mathbb {Z}_2\) . In this case, the quantity \(\nu _{2\pi /n}\)  can be abstractly defined as a quantity that takes the value "0" in the trivial phase and "1" in the topological phase. For weak topological phases, the classifying group can be isomorphic to \(\mathbb {Z}_2\)  or \(\mathbb {Z}\simeq 2\mathbb {Z}\) . In the latter case, the quantity \(\nu _{i}\)  should take the value "1" or "2" for the generator of the abelian group of topological equivalence classes of Hamiltonians and respect the abelian group property in its domain and image. Here, we distinguish the cases where the symmetries constrain a topological invariant to be even, see also the discussion in Sec. . This isomorphism is typically expressed for a periodic system as a quantity of the reciprocal space Hamiltonian, such as the Chern number [2]}, or a symmetry-based indicator [3]}. Explicit expressions for the topological invariants for topological superconductors in Cartan class D using symmetry-based indicators are presented in Appendix . In general, explicit expressions for topological invariants of higher-order topological phases can be derived using the method outlined in Refs. [4]}, [5]}.,
respectively, to the number of \((d-2)\) -dimensional anomalous states \(\theta _{n}^{\text{disc}}\)  at a disclination with rotation holonomy \(\Omega \)  and translation holonomy \(\mathbf {T}\)  are
\(\theta _2^\text{disc} & = \frac{\Omega }{\pi } \nu _\pi + \mathbf {T} \cdot \mathbf {G}_\nu \mod {2} \\\theta _3^\text{disc} & = T_z \nu _z \\\theta _4^\text{disc} & = \frac{2 \Omega }{\pi } \nu _{\pi / 2} + \mathbf {T} \cdot \mathbf {G}_\nu \mod {2} \\\theta _6^\text{disc} & = \frac{3 \Omega }{\pi } \nu _{\pi / 3} + T_z \nu _z\mod {2} .\) 
